Linear Recurrence Sequences Research Articles

Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

Higher Reciprocity Laws and Ternary Linear Recurrence Sequences

On the p-Adic Valuation of Third Order Linear Recurrence Sequences

On the p-Adic Valuation of Third Order Linear Recurrence Sequences

Kepler sets of second-order linear recurrence sequences over Qp

Let [Formula: see text] be a second-order linear recurrence sequence with constant coefficients over the field of [Formula: see text]-adic numbers [Formula: see text]. We study the set of limit points of the sequence of consecutive ratios [Formula: see text] in [Formula: see text].

Ruin Probabilities as Recurrence Sequences in a Discrete-Time Risk Process

The theory of linear recurrence sequences is applied to obtain an explicit formula for the ultimate ruin probability in a discrete-time risk process. It is assumed that the claims distribution is arbitrary but has finite support {0,1,…,m+1}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{\\{0,1,\\ldots ,m+1\\}}$$\\end{document}, for some integer m≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{m\\ge 1}$$\\end{document}. The method requires finding the zeroes of an m degree polynomial and solving a system of m linear equations. An approximation is derived and some numerical results and plots are provided as examples.

On multiplicative dependent vectors with coordinates from certain higher order recurrences

In this paper, we prove that there are finitely many multiplicative dependent vectors with coordinates from non-degenerate linear recurrence sequences [Formula: see text] of a fixed order [Formula: see text] These sequences satisfy the recurrence relation [Formula: see text] with initials [Formula: see text]. Here, the coefficients of the recurrence relation are positive integers satisfying [Formula: see text] or [Formula: see text] In both these conditions, [Formula: see text]

On Robustness for the Skolem, Positivity and Ultimate Positivity Problems

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our techniques also allow us to tackle robustness for ultimate positivity, which asks whether there is a bound on the number of steps after which the LRS remains positive. There are two variants depending on whether we ask for a "uniform" bound on this number of steps. For the non-uniform variant, when the neighbourhood is open, the problem turns out to be tractable, even when the neighbourhood is given as input.

The ring of finite algebraic numbers and its application to the law of decomposition of primes

In this paper, we develop an explicit method to express finite algebraic numbers (in particular, certain idempotents among them) in terms of linear recurrent sequences, and give applications to the characterization of the splitting primes in a given finite Galois extension over the rational field.

Вычисление тропических последовательностей, ассоциированных с последовательностями Сомоса, в пакете Gfan

This paper examines tropical recurrent sequences associated with Somos sequences. The classical Somos sequences have applications in the theory of elliptic curves. Due to the Laurent character of the classical sequences, a pattern can be inferred between the classical sequences and their tropical counterparts. The greatest interest is the increase in the dimension of the solution space of tropical sequences depending on the length of the final sequence. For a set of tropical sequences described by tropical recurrence relations, D.Yu.Grigoriev put forward a hypothesis about the stabilization of the maximum dimensions of the components of the corresponding tropical prevarieties. This hypothesis has been proven for tropical linear recurrent sequences. As part of this work, for tropical recurrent sequences associated with the sequences Somos-4 and Somos-5, the corresponding tropical prevarieties were investigated using the Gfan package in order to test Grigoriev's hypothesis.

Positivity-hardness results on Markov decision processes

This paper investigates a series of optimization problems for one-counter Markov decision processes (MDPs) and integer-weighted MDPs with finite state space. Specifically, it considers problems addressing termination probabilities and expected termination times for one-counter MDPs, as well as satisfaction probabilities of energy objectives, conditional and partial expectations, satisfaction probabilities of constraints on the total accumulated weight, the computation of quantiles for the accumulated weight, and the conditional value-at-risk for accumulated weights for integer-weighted MDPs. Although algorithmic results are available for some special instances, the decidability status of the decision versions of these problems is unknown in general. The paper demonstrates that these optimization problems are inherently mathematically difficult by providing polynomial-time reductions from the Positivity problem for linear recurrence sequences. This problem is a well-known number-theoretic problem whose decidability status has been open for decades and it is known that decidability of the Positivity problem would have far-reaching consequences in analytic number theory. So, the reductions presented in the paper show that an algorithmic solution to any of the investigated problems is not possible without a major breakthrough in analytic number theory. The reductions rely on the construction of MDP-gadgets that encode the initial values and linear recurrence relations of linear recurrence sequences. These gadgets can flexibly be adjusted to prove the various Positivity-hardness results.

Fibonacci primes, primes of the form 2n − k and beyond

We speculate on the distribution of primes in exponentially growing, linear recurrence sequences (un)n≥0 in the integers. By tweaking a heuristic which is successfully used to predict the number of prime values of polynomials, we guess that either there are only finitely many primes un, or else there exists a constant cu>0 (which we can give good approximations to) such that there are ∼culog⁡N primes un with n≤N, as N→∞. We compare our conjecture to the limited amount of data that we can compile. One new feature is that the primes in our Euler product are not taken in order of their size, but rather in order of the size of the period of the un(modp).

Greatest common divisors for polynomials in almost units and applications to linear recurrence sequences

Greatest common divisors for polynomials in almost units and applications to linear recurrence sequences

Exceptional biases in counting primes over function fields

AbstractWe study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call “complete bias,” “lower order bias,” and “reversed bias,” occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in as , a power of a fixed prime, goes to infinity. The bounds given extend a previous result of Kowalski, who studied a similar question along particular one‐parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret–Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry.

Skolem and positivity completeness of ergodic Markov chains

We consider the following Markov Reachability decision problems that view Markov Chains as Linear Dynamical Systems: given a finite, rational Markov Chain, source and target states, and a rational threshold, does the probability of reaching the target from the source at the nth step: (i) equal the threshold for some n? (ii) cross the threshold for some n? (iii) cross the threshold for infinitely many n? These problems are respectively known to be equivalent to the Skolem, Positivity, and Ultimate Positivity problems for Linear Recurrence Sequences (LRS), number-theoretic problems whose decidability has been open for decades. We present an elementary reduction from LRS Problems to Markov Reachability Problems that improves the state of the art as follows. (a) We map LRS to ergodic (irreducible and aperiodic) Markov Chains that are ubiquitous, not least by virtue of their spectral structure, and (b) our reduction maps LRS of order k to Markov Chains of order k+1: a substantial improvement over the previous reduction that mapped LRS of order k to reducible and periodic Markov chains of order 4k+5. This contribution is significant in view of the fact that the number-theoretic hardness of verifying Linear Dynamical Systems can often be mitigated by spectral assumptions and restrictions on order.

On products of prime powers in linear recurrence sequences

On products of prime powers in linear recurrence sequences

Computing the Density of the Positivity Set for Linear Recurrence Sequences

The set of indices that correspond to the positive entries of a sequence of numbers is called its positivity set. In this paper, we study the density of the positivity set of a given linear recurrence sequence, that is the question of how much more frequent are the positive entries compared to the non-positive ones. We show that one can compute this density to arbitrary precision, as well as decide whether it is equal to zero (or one). If the sequence is diagonalisable, we prove that its positivity set is finite if and only if its density is zero. Further, arithmetic properties of densities are treated, in particular we prove that it is decidable whether the density is a rational number, given that the recurrence sequence has at most one pair of dominant complex roots. Finally, we generalise all these results to symbolic orbits of linear dynamical systems, thereby showing that one can decide various properties of such systems, up to a set of density zero.

Some notes on the algebraic structure of linear recurrent sequences

AbstractSeveral operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products is an R-algebra, given any commutative ring R with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these R-algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.

The Solution of Fermat’s Two Squares Equation and Its Generalization In Lucas Sequences

As it is well known, there are an infinite number of primes in special forms such as Fermat's two squares form, p=x^2+y^2 or its generalization, p=x^2+y^4, where the unknowns x, y, and p represent integers. The main goal of this paper is to see if these forms still have an infinite number of solutions when the unknowns are derived from sequences with an infinite number of prime numbers in their terms. This paper focuses on the solutions to these forms where the unknowns represent terms in certain binary linear recurrence sequences known as the Lucas sequences of the first and second types.

Zero-Order Markov Processes with Multiple Final Sequences of States

A zero-order Markov process with multiple final sequences of states represents a stochastic system with independent transitions that stops its evolution as soon as one of the given final sequences of states is reached. The transition time of the system is unitary and the transition probability depends only on the destination state. It is proved that the distribution of the evolution time is a homogeneous linear recurrent sequence and a polynomial algorithm to determine the initial state and the generating vector of this recurrence is developed. Using the generating function, the main probabilistic characteristics are determined.

On the Sum of Digits of Expansions of a Pair of Consecutive Numbers over a Linear Recurrent Sequence

On the Sum of Digits of Expansions of a Pair of Consecutive Numbers over a Linear Recurrent Sequence

On Tribonacci functions and Gaussian Tribonacci functions

In this work, Gaussian Tribonacci functions are defined and investigated on the set of real numbers $\mathbb{R}$, i.e., functions $f_G: \mathbb{R} \rightarrow \mathbb{C}$ such that for all $x \in \mathbb{R}, n \in \mathbb{Z}, f_G(x+n)=$ $f(x+n)+i f(x+n-1)$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ is a Tribonacci function which is given as $f(x+3)=$ $f(x+2)+f(x+1)+f(x)$ for all $x \in \mathbb{R}$. Then the concept of Gaussian Tribonacci functions by using the concept of $f$-even and $f$-odd functions is developed. Also, we present linear sum formulas of Gaussian Tribonacci functions. Moreover, it is showed that if $f_G$ is a Gaussian Tribonacci function with Tribonacci function $f$, then $\lim _{x \rightarrow \infty} \frac{f_G(x+1)}{f_G(x)}=\alpha$ and $\lim _{x \rightarrow \infty} \frac{f_G(x)}{f(x)}=\alpha+i$, where $\alpha$ is the positive real root of equation $x^3-x^2-x-1=0$ for which $\alpha>1$. Finally, matrix formulations of Tribonacci functions and Gaussian Tribonacci functions are given.
 In the literature, there are several studies on the functions of linear recurrent sequences such as Fibonacci functions and Tribonacci functions. However, there are no study on Gaussian functions of linear recurrent sequences such as Gaussian Tribonacci and Gaussian Tetranacci functions and they are waiting for the investigating.
 We also present linear sum formulas and matrix formulations of Tribonacci functions which have not been studied in the literature.